RANK SUM TEST

Name:

RANK SUM TEST

Type:

Analysis Command

Purpose:

Perform a two sample rank sum test.

Description:

The t-test is the standard test for
testing that the difference between population means for two non-paired
samples are equal. If the populations are non-normal, particularly for
small samples, then the t-test may not be valid. The rank sum
test is an alternative that can be applied when distributional
assumptions are suspect. However, it is not as powerful as the
t-test when the distributional assumptions are in fact valid.

The rank sum test is also commonly called the Mann-Whitney rank sum
test or simply the Mann-Whitney test. Note that even though this test
is commonly called the Mann-Whitney test, it was in fact developed by
Wilcoxon.

To form the rank sum test, rank the combined samples. Then compute
the sum of the ranks for sample one, T1, and the sum of the
ranks for sample two, T2. If the sample sizes are equal,
the rank sum test statistic is the minimum of T1 and
T2. If the sample sizes are unequal, then find
T1 equal the sum of the ranks for the smaller sample. Then
compute T2 = n1(n1 +
n2 + 1) - T1. T is the minimum of
T1 and T2. Sufficiently small values of T cause
rejection of the null hypothesis that the sample means are equal.

Significance levels have been tabulated for small values of
n1 and n2. For sufficiently large
n1 and n2, the following normal
approximation is used:

\( Z = \frac{|\mu - T| - 0.5} {\sigma} \)

where

\( \mu = n_1(n_1 + n_2 + 1)/2 \)

\( \sigma = \sqrt{n_2 \mu /6} \)

Syntax:

RANK SUM TEST <y1> <y2>
<SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
<y2> is the second response variable;
and where the <SUBSET/EXCEPT/FOR qualification>
is optional.

Examples:

RANK SUM TEST Y1 Y2
RANK SUM TEST Y1 Y2 SUBSET TAG > 2

Note:

Dataplot saves the following internal parameters after a
rank sum test:

STATVAL

-

The rank sum test statistic

STATCD2

-

the normal cdf value of T (only applies for sufficiently
large N1 and N2)

CUTLOW90

-

0.05 critical value

CUTUPP90

-

0.95 critical value

CUTLOW95

-

0.025 critical value

CUTUPP95

-

0.975 critical value

CUTLOW99

-

0.005 critical value

CUTUPP99

-

0.995 critical value

Note that the above critical values are the lower and upper
tails for two sided tests (i.e., each tail is alpha/2. For
example, CUTLOW90 is the lower 5% of the normal percent point
function (adjusted for the mean and standard deviation). This
is the critical regions for alpha = 0.10, so there is 0.05 in
each tail.